We spent the month of January in Southern Arizona. We had been there before so knew where we wanted to go. We generally prefer camping in undeveloped locations but didn't specifically have that information. Because of the increase in camping activity, we made reservations in four state parks and found all to be nice places to be. Yep, more people than we'd like but had power and spaces are larger and more widely spaced than you'll find in commercial parks.
Our method was to go to a park, set up the 20 foot travel trailer and then explore daily. We ranged pretty widely and came back in the evenings or late afternoons.
Organ Pipe CG is nice (no power), Patagonia SP was great. we had a kayak and spend some of each day on the lake. Lots to see around Kartchner Caverns SP. Lost Dutchman is a beautiful park, but close to Phoenix. showers available at all except OrganPipe.
If you know remote sites you like, maybe a mixture of SP's and remote sites.
We finished the month with a night at the La Posada Inn in Winslow. If you aren't aware of it, check it out. rate was reasonable, restaurant is wonderful and the hotel is special as well... laposada.org